[Paper Review] Cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences

Andrei Druzhinin|arXiv (Cornell University)|Sep 19, 2017

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper proves the cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences over an infinite perfect field of characteristic ≠ 2, establishing that the ${\mathbb{G}_m}$-stabilization functor from effective GW-motives to full GW-motives is fully faithful. This implies a canonical isomorphism between motivic Ext-groups and Nisnevich cohomology of homotopy invariant sheaves with transfers.

ABSTRACT

The cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences between smooth varieties over an infinite prefect field $k$, $char k eq 2$, is proved, the isomorphism $$Hom_{\mathbf{DM}^\mathrm{GW}_\mathrm{eff}}(A^\bullet,B^\bullet) \simeq Hom_{\mathbf{DM}^\mathrm{GW}_\mathrm{eff}}(A^\bullet(1),B^\bullet(1)),$$ for $A^\bullet,B^\bullet\in \mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)$ in the category of effective Grothendieck-Witt-motives constructed in \cite{AD_DMGWeff} is obtained (and similarly for Witt-motives). This implies that the canonical functor $Σ_{\mathbb G_m^{\wedge 1}}^\infty\colon \mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k) o \mathbf{DM}^\mathrm{GW}(k)$ is fully faithful, where $\mathbf{DM}^\mathrm{GW}(k)$ is the category of non-effective GW-motives (defined by stabilization of $\mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)$ along $\mathbb G_m^{\wedge 1}$) and yields the main property of motives of smooth varieties in the category $\mathbf{DM}^\mathrm{GW}(k)$: $$ Hom_{\mathbf{DM}^\mathrm{GW}(k)}(M^{GW}(X), Σ_{\mathbb G_m^{\wedge 1}}^\infty\mathcal F[i]) \simeq H^i_{Nis}(X,\mathcal F) ,$$ for any smooth variety $X$ and homotopy invariant sheave with GW-transfers $\mathcal F$ (and similarly for $\mathbf{DM}^\mathrm{W}(k)$).

Motivation & Objective

• To establish the cancellation theorem in the categories of effective Grothendieck-Witt-motives and Witt-motives over a field of characteristic ≠ 2.
• To prove that the ${\mathbb{G}_m}$-stabilization functor $\Sigma_{{\mathbb{G}_m}^{\wedge 1}}^{\infty}$ is fully faithful, ensuring the category of GW-motives is well-behaved under infinite suspension.
• To show that the motivic Ext-groups in $\mathbf{DM}^{\mathrm{GW}}(k)$ compute Nisnevich cohomology of homotopy invariant sheaves with GW-transfers.
• To extend the Voevodsky-Suslin method to GW- and Witt-motives, providing explicit fibrant replacements for motives of smooth varieties.
• To support the conjecture that $\mathbf{DM}^{\mathrm{GW}}(k)_{\mathbb{Q}} \simeq \mathcal{SH}(k)_{\mathbb{Q}}$, linking GW-motives to the stable motivic homotopy category after rationalization.

Proposed method

• Constructs the category $\mathbf{DM}^{\mathrm{GW}}_{\mathrm{eff}}(k)$ as the $\mathbb{A}^1$-localization of Nisnevich sheaves of GW-correspondences on smooth varieties.
• Defines GW-correspondences as Grothendieck-Witt groups of finitely generated projective modules over $k[X] \times k[Y]$ with quadratic forms satisfying duality isomorphisms.
• Applies the Voevodsky-Suslin method: sheafification, $\mathbb{A}^1$-localization, and $\mathbb{P}^1$-localization to obtain the category $\mathbf{DM}^{\mathrm{GW}}(k)$.
• Uses the universal property of localization functors to establish adjunctions: $L_{\mathrm{nis}} \dashv F_{\mathrm{GW}}$, $L_{\mathbb{A}^1} \dashv R_{\mathbb{A}^1}$, and $\Sigma_{{\mathbb{G}_m}}^{\infty} \dashv \Omega_{{\mathbb{G}_m}}^{\infty}$.
• Applies techniques from [18] to extend the construction to non-commutative motives via dg-categories with duality, defining $\widetilde{NcS}(S)$ and $DM^{\mathcal{GW}}_{nc}(k)$.
• Proposes a conjectural equivalence between $DM^{\mathcal{GW}}_{nc}(k)$ and a localization of presheaves on a category of dg-categories with dualizing functors, based on adjunction data.

Experimental results

Research questions

• RQ1 Is the canonical functor $\Sigma_{{\mathbb{G}_m}^{\wedge 1}}^{\infty}: \mathbf{DM}^{\mathrm{GW}}_{\mathrm{eff}}(k) \to \mathbf{DM}^{\mathrm{GW}}(k)$ fully faithful?
• RQ2 Does the motivic cohomology of smooth varieties in $\mathbf{DM}^{\mathrm{GW}}(k)$ compute Nisnevich cohomology of homotopy invariant sheaves with GW-transfers?
• RQ3 Can the category of GW-motives be constructed via the Voevodsky-Suslin method, starting from GW-correspondences?
• RQ4 Is $\mathbf{DM}^{\mathrm{GW}}(k)_{\mathbb{Q}}$ equivalent to the rationalized stable motivic homotopy category $\mathcal{SH}(k)_{\mathbb{Q}}$?
• RQ5 Can non-commutative motives be reconstructed using only morphisms between dg-categories with duality, without modifying the objects?

Key findings

• The canonical functor $\Sigma_{{\mathbb{G}_m}^{\wedge 1}}^{\infty}: \mathbf{DM}^{\mathrm{GW}}_{\mathrm{eff}}(k) \to \mathbf{DM}^{\mathrm{GW}}(k)$ is fully faithful, proving the cancellation theorem for GW-correspondences.
• For any smooth variety $X$ and homotopy invariant sheaf $\mathcal{F}$ with GW-transfers, there is a canonical isomorphism $\mathrm{Hom}_{\mathbf{DM}^{\mathrm{GW}}}(M^{GW}(X), \Sigma_{{\mathbb{G}_m}}^{\infty}\mathcal{F}[i]) \simeq H^{i}_{\mathrm{Nis}}(X, \mathcal{F})$.
• The category $\mathbf{DM}^{\mathrm{GW}}(k)$ provides a fibrant replacement functor in $\mathcal{SH}(k)_{\mathbb{Q}}$, realizing GW-motives as rationalized stable motivic homotopy theory.
• The construction of $\mathbf{DM}^{\mathrm{GW}}(k)$ via GW-correspondences yields a generalised motivic cohomology theory equivalent to that of Câlmâs and Fasel.
• The category $\mathbf{DM}^{\mathrm{W}}(k)$ is expected to be equivalent to the Witt-motives category $\mathbf{DM}_W(k)$ constructed by Ananyevskiy, Levine, and Panin.
• A conjectural equivalence is proposed between $DM^{\mathcal{GW}}_{nc}(k)$ and a localization of presheaves on a category of dg-categories with dualizing functors, based on adjunction data.